Effective condition number and its applications

Consider the over-determined system Fx = b where ${{\bf F}\in\mathcal{R}^{m \times n}, m \geq n}$ and rank (F) = r ≤ n, the effective condition number is defined by ${{\rm Cond_{-}eff }= \frac {\|{\bf b}\|}{\sigma_r\|{\bf x}\|}}$ , where the singular values of F are given as σ _max = σ ₁ ≥ σ ₂ ≥ . . . ≥ σ _r > 0 and σ _r+1 = . . . = σ _n = 0. For the general perturbed system (A+ΔA)(x+Δx) = b+Δb involving both ΔA and Δb, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz’s problem by the collocation Trefftz methods (CTM). Motz’s problem is the benchmark of singularity problems. We choose the general particular solutions ${v_L = \sum\nolimits_{k=0}^L d_k (\frac {r}{R_p})^{k+\frac 12}}$ ${{\rm cos}(k +\frac 12)\theta}$ with a radius parameter R _p . The CTM is used to seek the coefficients d _i by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter R _p = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of R _p is misleading. Under the optimal choice R _p = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.     

Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Let ${\mathcal{B}}$ be a centrally symmetric convex polygon of ?² and ‖p?q‖ be the distance between two points p,q∈?² in the normed plane whose unit ball is ${\mathcal{B}}$ . For a set T of n points (terminals) in ?², a ${\mathcal{B}}$ -network on T is a network N(T)=(V,E) with the property that its edges are parallel to the directions of ${\mathcal{B}}$ and for every pair of terminals t _i and t _j , the network N(T) contains a shortest ${\mathcal{B}}$ -path between them, i.e., a path of length ‖t _i ?t _j ‖. A minimum ${\mathcal{B}}$ -network on T is a ${\mathcal{B}}$ -network of minimum possible length. The problem of finding minimum ${\mathcal{B}}$ -networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX’99) in the case when the unit ball ${\mathcal{B}}$ is a square (and hence the distance ‖p?q‖ is the l ₁ or the l _∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun (Symposium on Computational Geometry, pp. 393–402, 2009) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4, 3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum ${\mathcal{B}}$ -network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (Chepoi et al. in Theor. Comput. Sci. 390:56–69, 2008, and APPROX-RANDOM, pp. 40–51, 2005) and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.     

Complexity of Rational and Irrational Nash Equilibria

We introduce two new natural decision problems, denoted as ? RATIONAL NASH and ? IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ? RATIONAL NASH and ? IRRATIONAL NASH. Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. The problem NASH-EQUIVALENCE asks whether or not the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ? RATIONAL NASH. The problem NASH-REDUCTION asks whether or not there is a so called Nash reduction: a suitable map between corresponding strategy sets of players that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of NASH-REDUCTION that witnesses ? IRRATIONAL NASH. As our main result, we provide two distinct reductions to simultaneously show that (i) NASH-EQUIVALENCE is co- $\mathcal{NP}$ -hard and ? RATIONAL NASH is $\mathcal{NP}$ -hard, and (ii) NASH-REDUCTION and ? IRRATIONAL NASH are both $\mathcal{NP}$ -hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm (Proceedings of the 18th Joint Conference on Artificial Intelligence, pp. 765–771, 2003; Games Econ. Behav. 63(2), 621–641, 2008).     

Local noncuppability in R/M

Given any [c], [a], [d] xxxxxxxxR/M such that [d] ≤ [a] ≤ [c], [a] is locally noncuppable between [c] and [d] if [d] < [a] ≤ [c] and [a] ∨ [b] < [c] for any [b] xxxxxxxxR/M such that [d] ≤ [b] < [c]. It will be shown that given any nonzero [c] xxxxxxxxR/M, there are [a], [d] xxxxxxxxR/M such that [d] < [a] ≤ [c] and [a] is locally noncuppable between [c] and [d].     

Finding conclusion stability for selecting the best effort predictor in software effort estimation

Background: Conclusion Instability in software effort estimation (SEE) refers to the inconsistent results produced by a diversity of predictors using different datasets. This is largely due to the “ranking instability” problem, which is highly related to the evaluation criteria and the subset of the data being used. Aim: To determine stable rankings of different predictors. Method: 90 predictors are used with 20 datasets and evaluated using 7 performance measures, whose results are subject to Wilcoxon rank test (95 %). These results are called the “aggregate results”. The aggregate results are challenged by a sanity check, which focuses on a single error measure (MRE) and uses a newly developed evaluation algorithm called CLUSTER. These results are called the “specific results.” Results: Aggregate results show that: (1) It is now possible to draw stable conclusions about the relative performance of SEE predictors; (2) Regression trees or analogy-based methods are the best performers. The aggregate results are also confirmed by the specific results of the sanity check. Conclusion: This study offers means to address the conclusion instability issue in SEE, which is an important finding for empirical software engineering.     

PP-Lowness and a Simple Definition of AWPP

Abstract. We show that the counting classes AWPP and APP [FFKL], [L] are more robust than previously thought. Our results identify a sufficient condition for a language to be low for PP, and we show that this condition is at least as weak as other previously studied criteria. We extend a result of Köbler et al. by proving that all sparse co-C ₌ P languages are in APP, and are thus PP-low. Our results also imply that AWPP ?eq APP, and thus APP contains many other established subclasses of PP-low, thereby reducing several different lowness results to membership in APP. We also show that AWPP and APP are Σ ⁰ ₂ -definable classes. Some of our results are reminiscent of amplifying certainty in probabilistic computation.     

Algorithms for promoting anonymity of BS and for prolonging network lifetime of WSN

Two novel algorithms called AERO (aware of energy residue oriented) and free-AERO are proposed in this work. They are adopted in boosting the anonymity of a mobile base station (BS) and prolonging the overall lifetime of WSN (wireless sensor network). Without including in our proposed is the trivial one fixed-BS. Generally, BS play an important role of gathering data from all sensor nodes deployed over a wide WSN region. Thus, not only the BS location can be protected accordingly by the calculation of obtained data, but it makes more difficult for a malicious entity to violate the network. To complete the aforementioned objectives are the primary motivation of the proposed algorithms. On the other hand, the contribution of the proposed two algorithms is both going to boost BS anonymity and to prolong network lifetime for a WSN. Entropy and GSAT test are adopted as the criteria in system performance comparison for the deployment of AERO and free-AERO algorithms. Two scenarios with 3?×?3 and 5?×?5 cells are conducted in the simulation. Experimental results show that the proposed methods definitely boost BS anonymity and greatly prolong the network lifetime very well. Several results are illustrated and compared to previous studies for validation. When the scale of the nodes in the network becomes larger (i.e., as much as double), the method becomes more effective. The frequency of BS movement can also be adjusted with different values, 30 min, 60 min, or 90 min, to improve BS anonymity as needed.     

Bisection forAx=λBx with matrices of variable band width

Some extensions of the bisection method and of the inverse vector iteration for the general eigenvalue problemAx=λBx with symmetric matrices are given. A version with restricted pivoting is applied to sparse matricesA andB in which case the decomposition ofA-μB can be performed within an extended envelope with respect to the envelopeA andB. The effect of these refinements is illustrated by an example.     

Amplifying circuit lower bounds against polynomial time, with applications

We give a self-reduction for the Circuit Evaluation problem (CircEval) and prove the following consequences.

1. Amplifying size–depth lower bounds. If CircEval has Boolean circuits of n ^k size and n ^1?δ depth for some k and δ, then for every ${\epsilon > 0}$ , there is a δ′ > 0 such that CircEval has circuits of ${n^{1 + \epsilon}}$ size and ${n^{1- \delta^{\prime}}}$ depth. Moreover, the resulting circuits require only ${\tilde{O}(n^{\epsilon})}$ bits of non-uniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound).
2. Lower bounds for quantified Boolean formulas. Let c, d > 1 and e < 1 satisfy c < (1 ? e + d )/d. Either the problem of recognizing valid quantified Boolean formulas (QBF) is not solvable in TIME[n ^c ], or the Circuit Evaluation problem cannot be solved with circuits of n ^d size and n ^e depth. This implies unconditional polynomial-time uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n ^c -time uniform NC circuits, for all c < 2.

     

Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke counter-model for σ. Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.     

MPaaS: Mobility prediction as a service in telecom cloud

Mobile applications and services relying on mobility prediction have recently spurred lots of interest. In this paper, we propose mobility prediction based on cellular traces as an infrastructural level service of telecom cloud. Mobility Prediction as a Service (MPaaS) embeds mobility mining and forecasting algorithms into a cloud-based user location tracking framework. By empowering MPaaS, the hosted 3rd-party and value-added services can benefit from online mobility prediction. Particularly we took Mobility-aware Personalization and Predictive Resource Allocation as key features to elaborate how MPaaS drives new fashion of mobile cloud applications. Due to the randomness of human mobility patterns, mobility predicting remains a very challenging task in MPaaS research. Our preliminary study observed collective behavioral patterns (CBP) in mobility of crowds, and proposed a CBP-based mobility predictor. MPaaS system equips a hybrid predictor fusing both CBP-based scheme and Markov-based predictor to provide telecom cloud with large-scale mobility prediction capacity.     

A blind digital audio watermarking scheme based on EMD and UISA techniques

In this paper, a new perceptual spread spectrum audio watermarking scheme is discussed. The watermark embedding process is performed in the Empirical Mode Decomposition (EMD) domain, and the hybrid watermark extraction process is based on the combination of EMD and ISA (Independent Subspace Analysis) techniques, followed by the generic detection system, i.e. inverse perceptual filter, predictor filter and correlation based detector. Since the EMD decomposes the audio signal into several oscillating components–the intrinsic mode functions (IMF)–the watermark information can be inserted in more than one IMF, using spread spectrum modulation, allowing hence the increase of the insertion capacity. The imperceptibility of the inserted data is ensured by the use of a psychoacoustical model. The blind extraction of the watermark signal, from the received watermarked audio, consists in the separation of the watermark from the IMFs of the received audio signal. The separation is achieved by a new proposed under-determined ISA method, here referred to as UISA. The proposed hybrid watermarking system was applied to the SQAM (Sound Quality Assessment Material) audio database (Available at http://sound.media.mit.edu/mpeg4/audio/sqam/) and proved to have efficient detection performances in terms of Bit Error Rate (BER) compared to a generic perceptual spread spectrum watermarking system. The perceptual quality of the watermarked audio was objectively assessed using the PEMO-Q (Tool for objective perceptual assessment of audio quality) algorithm. Also, using our technique, we can extract the different watermarks without using any information of original signal or the inserted watermark. Experimental results exhibit that the transparency and high robustness of the watermarked audio can be achieved simultaneously with a substantial increase of the amount of information transmitted. A reliability of 1.8 10^???4 (against 1.5 10^???2 for the generic system), for a bit rate of 400 bits/s, can be achieved when the channel is not disturbed.     

Query-Efficient Locally Decodable Codes of Subexponential Length

A k-query locally decodable code (LDC) C : Σ ⁿ → Γ ^N encodes each message x into a codeword C(x) such that each symbol of x can be probabilistically recovered by querying only k coordinates of C(x), even after a constant fraction of the coordinates has been corrupted. Yekhanin (in J ACM 55:1–16, 2008) constructed a 3-query LDC of subexponential length, N = exp(exp(O(log n/log log n))), under the assumption that there are infinitely many Mersenne primes. Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009) constructed a 3-query LDC of length ${N_{2}={\rm exp}({\rm exp} (O(\sqrt{\log n\log\log n})))}$ with no assumption, and a 2 ^r -query LDC of length ${N_{r}={\rm exp}({\rm exp}(O(\sqrt[r]{\log n(\log \log n)^{r-1}})))}$ , for every integer r ≥ 2. Itoh and Suzuki (in IEICE Trans Inform Syst E93-D 2:263–270, 2010) gave a composition method in Efremenko’s framework and constructed a 3 · 2 ^r-2-query LDC of length N _r , for every integer r ≥ 4, which improved the query complexity of Efremenko’s LDC of the same length by a factor of 3/4. The main ingredient of Efremenko’s construction is the Grolmusz construction for super-polynomial size set-systems with restricted intersections, over ${\mathbb{Z}_m}$ , where m possesses a certain “good” algebraic property (related to the “algebraic niceness” property of Yekhanin in J ACM 55:1–16, 2008). Efremenko constructed a 3-query LDC based on m = 511 and left as an open problem to find other numbers that offer the same property for LDC constructions. In this paper, we develop the algebraic theory behind the constructions of Yekhanin (in J ACM 55:1–16, 2008) and Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009), in an attempt to understand the “algebraic niceness” phenomenon in ${\mathbb{Z}_m}$ . We show that every integer m =  pq = 2 ^t ?1, where p, q, and t are prime, possesses the same good algebraic property as m = 511 that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki’s composition method. More precisely, we construct a ${3^{\lceil r/2\rceil}}$ -query LDC for every positive integer r < 104 and a ${\left\lfloor (3/4)^{51} \cdot 2^{r}\right\rfloor}$ -query LDC for every integer r ≥ 104, both of length N _r , improving the 2 ^r queries used by Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009) and 3 · 2 ^r-2 queries used by Itoh and Suzuki (in IEICE Trans Inform Syst E93-D 2:263–270, 2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.     

Iterations for solving linear variational inequalities on domains with nonlinear boundaries

This paper deals with numerical methods for solving linear variational inequalities on an arbitrary closed convex subsetC of ? ⁿ . Although there were numerous iterations studied for the caseC=? ₊ ⁿ , few were proposed for the case whenC is a closed convex subset. The essential difficulty in this case is the nonlinearities ofC's boundaries. In this paper iteration processes are designed for solving linear variational inequalities on an arbitrary closed convex subsetC. In our algorithms the computation of a linear variational inequality is decomposed into a sequence of problems of projecting a vector to the closed convex subsetC, which are computable as long as the equations describing the boundaries are given. In particular, using our iterations one can easily compute a solution whenC is one of the common closed convex subsets such as cube, ball, ellipsoid, etc. The non-accurate iteration, the estimate of the solutions on unbounded domains and the theory of approximating the boundaries are also established. Moreover, a necessary and sufficient condition is given for a vector to be an approximate solution. Finally, some numerical examples are presented, which show that the designed algorithms are effective and efficient. The exposition of this paper is self-contained.     

A new method for solving an arbitrary fully fuzzy linear system

In this paper, We propose a simple and practical method (that works only for triangular fuzzy numbers) to solve an arbitrary fully fuzzy linear system (FFLS) in the form $\widetilde{A}\otimes \widetilde{x}=\widetilde{b},$ where $\widetilde{A}_{n \times n}$ is a fuzzy matrix, $\widetilde{x}$ and $\widetilde{b}$ are n × 1 fuzzy vectors. The idea of the presented method is constructed based on the extending 0-cut and 1-cut solution of original fully fuzzy linear systems (FFLS). We also define a fuzzy solution of FFLS and establish the necessary and sufficient conditions for the uniqueness of a fuzzy solution.     

Simulation of an n^{+}\hbox {-}n\hbox {-}n^{+} Diode by Using Globally-Hyperbolically-Closed High-Order Moment Models

By the moment closure of the Boltzmann transport equation, the extended hydrodynamic models for electron transport have been derived in Cai et al. (J Math Phys 53:103503, 2012). With the numerical scheme developed in Li et al. (2012) recently, it has been demonstrated that the derived extended hydrodynamic models can capture the major features of the solution of kinetic equations. As the application of the models and the numerical scheme proposed therein, we in this paper carry out the numerical simulation to investigate the carrier transport in $n^{+}$ - $n$ - $n^{+}$ structures by solving the moment system derived from the Boltzmann–Poisson equations. Without any additional empirical parameters than that used in directly solving the kinetic equations, we obtain numerical results by the extended hydrodynamic models with very satisfied agreement with the solution of the kinetic equations, even in case that the length of the channel is as short as 50 nm.     

Vertex Cover Kernelization Revisited

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G′,k′) in polynomial time with the guarantee that G′ has at most 2k′ vertices (and thus $\mathcal{O}((k')^{2})$ edges) with k′≤k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Θ(k ²) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number $\mathop{\mathrm{\mbox {\textsc{vc}}}}(G)$ since $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\leq\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)$ and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ : an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G′,X′,k′) such that |V(G′)|≤2k and $|V(G')| \in\mathcal{O}(|X'|^{3})$ . A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP ? coNP/poly and the polynomial hierarchy collapses to the third level.     

Query reverse engineering

In this paper, we introduce a new problem termed query reverse engineering (QRE). Given a database \(D\) and a result table \(T\) —the output of some known or unknown query \(Q\) on \(D\) —the goal of QRE is to reverse-engineer a query \(Q'\) such that the output of query \(Q'\) on database \(D\) (denoted by \(Q'(D)\) ) is equal to \(T\) (i.e., \(Q(D)\) ). The QRE problem has useful applications in database usability, data analysis, and data security. In this work, we propose a data-driven approach, TALOS for Tree-based classifier with At Least One Semantics, that is based on a novel dynamic data classification formulation and extend the approach to efficiently support the three key dimensions of the QRE problem: whether the input query is known/unknown, supporting different query fragments, and supporting multiple database versions.     

Reasoning about temporal properties of rational play

This article is about defining a suitable logic for expressing classical game theoretical notions. We define an extension of alternating-time temporal logic (ATL) that enables us to express various rationality assumptions of intelligent agents. Our proposal, the logic ATLP (ATL with plausibility) allows us to specify sets of rational strategy profiles in the object language, and reason about agents’ play if only these strategy profiles were allowed. For example, we may assume the agents to play only Nash equilibria, Pareto-optimal profiles or undominated strategies, and ask about the resulting behaviour (and outcomes) under such an assumption. The logic also gives rise to generalized versions of classical solution concepts through characterizing patterns of payoffs by suitably parameterized formulae of ATLP. We investigate the complexity of model checking ATLP for several classes of formulae: It ranges from $\Delta_{\mathbf{3}}^{\mathbf{P}}$ to PSPACE in the general case and from $\Delta_{\mathbf{3}}^{\mathbf{P}}$ to $\Delta_{\mathbf{4}}^{\mathbf{P}}$ for the most interesting subclasses, and roughly corresponds to solving extensive games with imperfect information.     

An SDP Primal-Dual Algorithm for Approximating the Lovász-Theta Function

The Lovász ?-function (Lovász in IEEE Trans. Inf. Theory, 25:1–7, 1979) of a graph G=(V,E) can be defined as the maximum of the sum of the entries of a positive semidefinite matrix X, whose trace Tr(X) equals 1, and X _ij =0 whenever {i,j}∈E. This function appears as a subroutine for many algorithms for graph problems such as maximum independent set and maximum clique. We apply Arora and Kale’s primal-dual method for SDP to design an algorithm to approximate the ?-function within an additive error of δ>0, which runs in time $O(\frac{\vartheta ^{2} n^{2}}{\delta^{2}} \log n \cdot M_{e})$ , where ?=?(G) and M _e =O(n ³) is the time for a matrix exponentiation operation. It follows that for perfect graphs G, our primal-dual method computes ?(G) exactly in time O(? ² n ⁵logn). Moreover, our techniques generalize to the weighted Lovász ?-function, and both the maximum independent set weight and the maximum clique weight for vertex weighted perfect graphs can be approximated within a factor of (1+?) in time O(? ^?2 n ⁵logn).